38edo

← 37edo

38edo

39edo →

Prime factorization	2 × 19
Step size	31.5789 ¢
Fifth	22\38 (694.737 ¢) (→ 11\19)
Semitones (A1:m2)	2:4 (63.16 ¢ : 126.3 ¢)
Consistency limit	5
Distinct consistency limit	5

Theory

Since 38 = 2*19, it can be thought of as two parallel 19edos. While the halving of the step size lowers consistency and leaves it only mediocre in terms of overall relative error, the fact that the 3rd & 5th harmonics are flat by almost exactly the same amount, while the 11th is double that means there are quite a few near perfect composite ratios, such as the the 6/5 it shares with 19edo, plus 11/9, 15/11 & 25/22, (and their inversions) while a single step nears 55/54. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30. It tempers out the same 5-limit commas as 19edo, namely 81/80, 3125/3072 and 15625/15552. In the 7-limit, we can add 50/49, and tempering out 81/80 and 50/49 gives injera temperament, for which 38 is the optimal patent val. In the 11-limit, we can add 121/120 and 176/175.

Approximation of prime harmonics in 38edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.0	-7.2	-7.4	+10.1	-14.5	+12.1	-10.2	-13.3	+3.3	+12.5	-8.2
	Relative (%)	+0.0	-22.9	-23.3	+32.1	-45.8	+38.3	-32.4	-42.1	+10.5	+39.7	-25.9
Steps (reduced)		38 (0)	60 (22)	88 (12)	107 (31)	131 (17)	141 (27)	155 (3)	161 (9)	172 (20)	185 (33)	188 (36)

Intervals

Step	Cents	21-limit ratios, treated as 38df	21-limit ratios, by direct mapping	Ups and Downs Notation*
0	0.0000			Perfect 1sn	P1	D
1	31.5789			Up 1sn	^1	^D
2	63.1579			Aug 1sn, dim 2nd	A1, d2	D#
3	94.7368			Upaug 1sn, downminor 2nd	^A1, vm2	^D#, vEb
4	126.3157			Minor 2nd	m2	Eb
5	157.8947			Mid 2nd	~2	vE
6	189.4737			Major 2nd	M2	E
7	221.0526			Upmajor 2nd	^M2	^E
8	252.6316			Aug 2nd, Dim 3rd	A2, d3	E#, Fb
9	284.2105			Downminor 3rd	vm3	vF
10	315.7895			Minor 3rd	m3	F
11	347.3684			Mid 3rd	~3	^F
12	378.9474			Major 3rd	M3	F#
13	410.5263			Upmajor 3rd, Downdim 4th	^M3, vd4	^F#, vGb
14	442.1053			Aug 3rd, dim 4th	A3, d4	Gb
15	473.6843			Down 4th	v4	vG
16	505.2632			Perfect 4th	P4	G
17	536.8421			Up 4th	^4	^G
18	568.4211			Aug 4th	A4	G#
19	600.0000			Upaug 4th, downdim 5th	^A4, vd5	^G#, vAb
20	631.5789			Dim 5th	d5	Ab
21	663.1579			Down 5th	v5	vA
22	694.7368			Perfect 5th	P5	A
23	726.3157			Up 5th	^5	^A
24	757.8947			Aug 5th, dim 6th	A5, d6	A#
25	789.4737			Upaug 5th, downminor 6th	^A5, vm6	^A#, vBb
26	821.0526			Minor 6th	m6	Bb
27	852.6316			Mid 6th	~6	vB
28	884.2105			Major 6th	M6	B
29	915.7895			Upmajor 6th	^M6	^B
30	947.3684			Aug 6th, dim 7th	A6, d7	B#, Cb
31	978.9474			Downminor 7th	vm7	vC
32	1010.5263			Minor 7th	m7	C
33	1042.1053			Mid 7th	~7	^C
34	1073.6843			Major 7th	M7	C#
35	1105.2632			Upmajor 7th, Downdim 8ve	^M7, vd8	^C#, vDb
36	1136.8421			Aug 7th, dim 8ve	A7, d8	Db
37	1168.4211			Down 8ve	v8	vD
38	1200.0000			Perfect 8ve	P8	D

* Ups and downs may be substituted with semi-sharps and semi-flats, respectively

Instruments

• Lumatone mapping for 38edo
• Skip fretting system 38 2 11

Music

• Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin by Claudi Meneghin